Pairs of dual Gabor frame generators with compact support and desired frequency localization

Abstract

Let g ∈ L 2 ( R ) be a compactly supported function, whose integer-translates { T k g } k ∈ Z form a partition of unity. We prove that for certain translation and modulation parameters, such a function g generates a Gabor frame, with a (noncanonical) dual generated by a finite linear combination h of the functions { T k g } k ∈ Z ; the coefficients in the linear combination are given explicitly. Thus, h has compact support and the decay in frequency is controlled by the decay of g ˆ . In particular, the result allows the construction of dual pairs of Gabor frames, where both generators are given explicitly, have compact support, and decay fast in the Fourier domain. We further relate the construction to wavelet theory. Letting D denote the dilation operator and B N be the Nth order B-spline, our results imply that there exist dual Gabor frames with generators of the type g = ∑ c k D T k B N and h = ∑ c ˜ k D T k B N , where both sums are finite. It is known that for N > 1 , such functions cannot generate dual wavelet frames { D j T k g } j , k ∈ Z , { D j T k h } j , k ∈ Z .